Let us prove part (REF ). Arguing as in [1]}, it can be proved that, for any \((G,D,\rho )\in \operatorname{\mathcal {QD}iv}_{\overline{G}}\) , the restriction map
\((\alpha _{\overline{\Gamma }})_{|P(G,D, \rho )}\colon P(G,D,\rho )\longrightarrow \operatorname{Pic}^{\deg (D)}(\overline{\Gamma })[A^{\deg (D)}_{\overline{\Gamma }}]{\cong }\operatorname{Jac}(\overline{\Gamma }).\) 
is affine, and hence continuous. Since \(\operatorname{Jac}_{\overline{\Gamma }}\)  is defined as the colimit of the polytopes \(P(G,D, \rho )\) , as \((G,D,\rho )\)  varies in \(\operatorname{\mathcal {QD}iv}_{\overline{G}}\) , it follows that \(\alpha _{\overline{\Gamma }}\)  is continuous. The map \(\alpha _{\overline{\Gamma }}\)  preserves the degree since \(D(v)=1\)  for every \(v\in V_{\operatorname{exc}}(G)\) . Finally, the fact that \(\alpha _{\overline{\Gamma }}\)  is surjective follows from the fact that every divisor \(D\)  on \(\overline{\Gamma }\)  is linearly equivalent to an admissible divisor on a quasistable model of \(\Gamma \) .
Part (REF ): the map \(\alpha _{\overline{\Gamma }}(\phi )\)  is continuous by part (REF ) and it is bijective by [1]} (also see the sequel, where we generalize this), and it is a homeomorphism since \(\operatorname{Jac}_{\overline{\Gamma }}(\phi )\)  is compact (being a colimit of polytopes) and \(\operatorname{Pic}^{|\phi |}(\overline{\Gamma })\)  is Hausdorff (being a real torus).
